Related papers: Symplectic embeddings of polydisks

We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible…

This note describes a correct way to perform the inflation procedures claimed in the papers on embedding ellipsoids, Journ. Top. 2 (2009), 1-22 and 589-623. The idea is to inflate along a collection of transversally and positively…

In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded…

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of…

This paper establishes robust obstructions to representing Hamiltonian diffeomorphisms as $k$-th powers ($k \geq 2$) or embedding them in flows for certain higher-dimensional symplectic manifolds $(M,\omega)$, including surface bundles. We…

An influential result of McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many…

An embedding $\varphi \colon (M_1, \omega_1) \to (M_2, \omega_2)$ (of symplectic manifolds of the same dimension) is called $\epsilon$-symplectic if the difference $\varphi^* \omega_2 - \omega_1$ is $\epsilon$-small with respect to a fixed…

Consider the $2n$-dimensional closed ball $B$ of radius 1 in the $2n$-dimensional symplectic cylinder $Z = D \times R^{2n-2}$ over the closed disc $D$ of radius 1. We construct for each $\epsilon >0$ a Hamiltonian deformation $\phi$ of $B$…

This paper continues the study of the ellipsoid embedding function of symplectic Hirzebruch surfaces parametrized by $b \in (0,1)$, the size of the symplectic blow-up. Cristofaro-Gardiner, et al. (arxiv: 2004.13062) found that if the…

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a…

We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $R^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The…

The ellipsoidal capacity function of a symplectic four manifold $X$ measures how much the form on $X$ must be dilated in order for it to admit an embedded ellipsoid of eccentricity $z$. In most cases there are just finitely many…

We consider a fibered Lagrangian $L$ in a compact symplectic fibration with small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks with Lagrangian boundary from the base to the total space. In case $L$ is a product,…

The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a $3$-dimensional space of constant curvature was settled by B\"or\"oczky and Florian for the hyperbolic space $\HYP$ in \cite{BF64} and by proving…

We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using symplectic field theory. We then compute…

We study symplectic structures on four-dimensional small covers. Our main result shows that every symplectic four-dimensional small cover is aspherical. We then classify symplectic small covers over products of two polygons, proving that…

We explicitly construct several Poisson structures with compact support. For example, we show that any Poisson structure on $\R^n$ with polynomial coefficients of degree at most two can be modified outside an open ball, such that it becomes…

In this paper, we study confoliations in dimensions higher than three mostly from the perspective of symplectic fillability. Our main result is that Massot-Niederkr\"uger-Wendl's bordered Legendrian open book, an object that obstructs the…

Let (M,\omega) be a symplectic manifold, and (\Sigma,\sigma) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {\omega^{D}}_{(\Sigma, \sigma)} on the space of immersions \Sigma \to M that is a special case of…

We show that all closed symplectic 4-manifolds have the packing stability property: there are no obstructions beyond volume to embedding a collection of sufficiently small balls. This generalizes a theorem of Biran which gives the same…